1999 Fiscal Year Final Research Report Summary
Differential equations on homogeneous spaces
| Project/Area Number |
09440048
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
解析学
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| Research Institution | University of Tokyo |
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Principal Investigator |
OSHIMA Toshio Graduate school of Mathematical Sciences, Univ. of Tokyo, Professor, 大学院・数理科学研究科, 教授 (50011721)
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| Co-Investigator(Kenkyū-buntansha) |
TERADA Itaru Graduate school of Mathematical Sciences, Univ. of Tokyo, Ass. Prof., 大学院・数理科学研究科, 助教授 (70180081)
MATSUMOTO Hisayoshi Graduate school of Mathematical Sciences, Univ. of Tokyo, Ass. Prof., 大学院・数理科学研究科, 助教授 (50272597)
ODA Takayuki Graduate school of Mathematical Sciences, Univ. of Tokyo, Professor, 大学院・数理科学研究科, 教授 (10109415)
SHIMENO Nobukazu Graduate school of Mathematical Sciences, Univ. of Tokyo, Lecturer, 理学部, 講師 (60254140)
KOBAYASHI Toshiyuki Graduate school of Mathematical Sciences, Univ. of Tokyo, Ass. Prof., 大学院・数理科学研究科, 助教授 (80201490)
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| Project Period (FY) |
1997 – 1999
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| Keywords | homogenneous space / representation theory / invariant differential equations / spherical function / integrable system / Poisson transform / semisimple Lie group / hypergeometric function |
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| Research Abstract |
1. We generalized the classical Capelli identities to the case of minors and proved that the operators defined by the identities give the generators of the system of differential equations which characterizes the degenerate series of GL(n). Moreover we studied differential operators in the form of poweres of matrices of Lie algebra and generators of the system of differential equations characterizing any degenerate series of classical Lie groups. We proved that the equations characterize the image of Poisson transforms of the space of functions on various boundaries of classical Lie groups.
2. We defined general hypergeometric function on a real semisimple Lie group G. Using this Capelli identities, we studied intertwining operators between degenerate series of GL(n) and proved that when they correspond to Radon transforms on real Grassmannians, they give generalization of Gelfand's hypergeometric functions and their natural interpretation form the vie point of representation theory. We also characterized the image of Radon transformation on real Grassmaninans in terms of differential equations.
3. We studied the multiplicity of the representation of a real semisimple Lie group G contained in a induced representation of G from a representation of a closed subgroup H of G. We obtained its good estimate from both sides and got a necessary and sufficient geometrical condition for (G, H) so that the multiplicity is finite or uniformly bounded. Here we assume the reductibity of H in the case when the dimension of the representation of H is infinite. For example, if G = U(p + 1, q) and H = U(p, q), the multiplicity of any holomorphic discrete series representation of G in the represenation induced from any irreducible admissible representaion of H is at most one.
4. We constructed all the higer integrals of the completely integrable quantum system invariant under the action of a classical Weyl group.
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